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Auction Game

In a common-value auction, the highest bidder “wins,” but makes a net loss unless all bids are so strongly risk-averse or “shaded” that even his own bid falls below the value of the purchased object once that value is objectively assessed (e.g., once the money in the jar is counted or the oil field has been fully explored). The winner’s curse depends on uncertainty and the unreliability of the value estimates at the time of auction. When uncertainty is high and the number of bidders is large, the winner’s curse is almost impossible to avoid.

If no one bids, no one will suffer the curse, but nor will anyone steal the item for cheap. This is the gate to the dilemma. Each individual has an incentive to bid. Yet, as the number of bids increases, the probability that the highest bid is too high, and the margin by which it is too high, also increase. At the same time, the probability of placing the highest bid decreases for each individual.

Not bidding can be seen as an act of cooperation. If no one bids, the status quo is preserved and no one gains or loses. Bidding is an act of defection, which promises a gain at first, when few bid, but all but guarantees a steep loss to the “winner” when many bid. The common-value auction is a game in which the most aggressive defector ultimately suckers himself.

We can translate these considerations into a payoff or preference matrix. In the simplest case, there are two players. Row player Al’s ranked preferences are shown in the Figure. Al neither gains nor loses if he rests [R] regardless of what Bud (column) does. In the top row of the matrix, this is indicated by the 0s. If Bud rests, however, Al prefers to bid [B] to realize a gain [1]. In contrast, if Bud bids [B], Al would be better off resting. If he were to bid, his chances of winning would be correlated with the height of his bid and hence with the probability and the size of the winner’s curse [-1].

The common-auction game has the same preference structure as the volunteer’s dilemma (VOD; Diekmann, 1985). In a 2-person VOD too, you (Al) are better off doing what the other person (Bud) is not doing. If you defect while the other volunteers, you win [1]. If both defect, the outcome is bad for both [-1]. If you volunteer, you reap a small psychic benefit [0 > -1] regardless of what the other does.

In the auction game, not bidding is thus analogous to volunteering. If the other player (all other players) rests, the single bidder has the advantage. If the other (all others) bids, only the “winner” will suffer a loss. This is where the auction game differs from the VOD. In the VOD, everyone suffers if everyone defects. In the auction game, there might be one winner if all others rest; in the VOD, there are many beneficiaries if only one player makes the sacrifice of volunteering. The result of this difference is that the temptation to defect is greater in the auction game. Only the VOD has the potential for collective disaster. And by the way, there is (almost) always one big winner in the auction game, namely the seller.

What is one to do? If the auction game is basically a type of VOD, then game theory can tells us what the best strategy is. Clearly, there is no dominating strategy. Each player wants to find a way to not duplicate the other player’s choice. This calls for a mixed strategy. In the VOD, one can derive the probability of volunteering at which the player maximizes his expected outcomes. In the 2-person game described above, a player could always volunteer or randomly defect half the time, and it wouldn’t matter. If, however, more players were added, it would be rational to increase one’s probability of defection. Likewise, it would be smart to defect more often if the payoff for unilateral defection were to become larger or the payoff for unanimous defection were to become smaller. The same considerations apply, mutatis mutandis, to the auction dilemma.

Because finding the optimal solution is difficult, it is also difficult to pin down outright irrationality. Thaler (1988) concludes that “The winner's curse is a prototype for the type of problem that is amenable to investigation using modern behavioral economics, a combination of cognitive psychology and microeconomics. The key ingredient is the existence of a cognitive illusion, a mental task that induces a substantial majority of subjects to make a systematic error” (p. 201). Notice that Thaler does not impugn specific choices or strategies of making choices [although in his paper he suggests that any bidding is irrational or that submitting a bid corresponding to one’s value estimate is irrational]. Instead, Thaler infers the existence of a cognitive illusion from the existence of an error, an error that is only revealed by the outcome of the decision. Other theorists (cf. Hastie & Dawes, 2010) call this outcome bias, which is widely regarded as a cognitive illusion.